Semiflow selection and Markov selection theorems

The deterministic analog of the Markov property of a time-homogeneous Markov process is the semigroup property of solutions of an autonomous differential equation. The semigroup property arises naturally when the solutions of a differential equation are unique, and leads to a semiflow. We prove an abstract result on measurable selection of a semiflow for the situations without uniqueness. We outline applications to ODEs, PDEs, differential inclusions, etc. Our proof of the semiflow selection theorem is motivated by N. V. Krylov's Markov selection theorem. To accentuate this connection, we include a new version of the Markov selection theorem related to more recent papers of Flandoli & Romito and Goldys et al.Comment: In this revised version we have added a new abstract result in Sec. 2. It is used to correct the Navier-Stokes example in application

Short Cycles in Repeated Exponentiation Modulo a Prime

Given a prime $p$, we consider the dynamical system generated by repeated exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le p-1$. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system

Wonderful blowups associated to group actions

A group action on a smooth variety provides it with the natural stratification by irreducible components of the fixed point sets of arbitrary subgroups. We show that the corresponding maximal wonderful blowup in the sense of MacPherson-Procesi has only abelian stabilizers. The result is inspired by the abelianization algorithm of Batyrev.Comment: 6 page

Edge Routing with Ordered Bundles

Edge bundling reduces the visual clutter in a drawing of a graph by uniting the edges into bundles. We propose a method of edge bundling drawing each edge of a bundle separately as in metro-maps and call our method ordered bundles. To produce aesthetically looking edge routes it minimizes a cost function on the edges. The cost function depends on the ink, required to draw the edges, the edge lengths, widths and separations. The cost also penalizes for too many edges passing through narrow channels by using the constrained Delaunay triangulation. The method avoids unnecessary edge-node and edge-edge crossings. To draw edges with the minimal number of crossings and separately within the same bundle we develop an efficient algorithm solving a variant of the metro-line crossing minimization problem. In general, the method creates clear and smooth edge routes giving an overview of the global graph structure, while still drawing each edge separately and thus enabling local analysis

Integration of fiber coupled high-Q silicon nitride microdisks with atom chips

Micron scale silicon nitride (SiN_x) microdisk optical resonators are demonstrated with Q = 3.6 x 10^6 and an effective mode volume of 15 (\lambda / n)^3 at near visible wavelengths. A hydrofluoric acid wet etch provides sensitive tuning of the microdisk resonances, and robust mounting of a fiber taper provides efficient fiber optic coupling to the microdisks while allowing unfettered optical access for laser cooling and trapping of atoms. Measurements indicate that cesium adsorption on the SiN_x surfaces significantly red-detunes the microdisk resonances. A technique for parallel integration of multiple (10) microdisks with a single fiber taper is also demonstrated.Comment: Published vesion. Minor change